Whitney's Umbrella

An interesting surface with intersecting polygons. The parameters are so simple.

They are:

x = u
y = v * v
z = u * v

with

u_start = -1
u_end = 1
v_start = -1
v_end = 1

The ICE Tree and parametric surface is as follow.

Little Update, I am working on the next set of ice surface posts. Here is a preview of the braided torus for your enjoyment!

And no, this is not the torus wireframe, this is the actual surface!

Great stuff. I hope to get this working in the near future after I finish a couple of projects I'm currently working on. Haven't had the chance to really take a look at ICE Modelling yet.

I can't wait till you generate 3d Mandelbulb!!
http://www.skytopia.com/project/fractal/mandelbulb.html

Great stuff - keep up the hard work!
-Dave

Hi Dave,

Thank you for your kind encouragement.



For that, you will need some sort of ray marcher / marching cube / voxelizer method ....
maybe emPolygoniser can help!

Maybe something to propose in the community project thread! That would be an interesting project!

Little update!

Just created a serie of supershapes based on the super torus.

Here is a rendered preview.

The SuperCone, SuperSphere, SuperCylinder, SuperDisk and SuperTorus!

Each shape can adopt many variations by adjusting its parameters! As an example, the rounded cube in the middle is actually the SuperSphere!

Very nice indeed! I've fallen behind in trying it myself (have to get some work done) but really admire what you've achieved here. Looking forward to more!

I have been experimenting with the super formula lately. Here are some Super Shapes based on Paul Bourke parameters on his website!

http://paulbourke.net/geometry/supershape3d/

All the shapes are created with the same compound by adjusting its parameters. You can get some very cool animation too!

Enjoy!

SuperShape Compound

I have been busy at the studio lately and I have to prepare a portrait class, so in the meantime, i'll give you the supershape compound to play with. It is based on the sphere formula but it does not have to be.

It's not pretty, I have not done the PPG Logic yet, but it can show you the power of this compound.

There is still some work to do to improve it (better layout, the superformula break at high enough numbers) but we can create so many shapes with it that it is worth playing with, even at it's early stage.

Inside, you will find my early version of the super formula. Copy this node and put it in an other polar coordinate shape like the disk, cylinder or cone and see what you can do with it. It's not called the super formula for nothing!

So have fun, check Paul Bourke website and other sites that have posted super shapes settings and see what you can reproduce.

This compound is fun to play with and animate! Enjoy!

The SuperFormula

The superformula is well explained on Paul Bourke website and on Wikipedia here:
http://en.wikipedia.org/wiki/Superformula

In polar coordinates, with r the radius and φ the angle, the superformula is as per the first figure below from Wikipedia.

The formula appeared in a work by Johan Gielis in 2003. It was obtained by generalizing the superellipse in order to create organic shapes. The superformula can be used to describe many complex shapes and curves that are found in nature by inserting the formula into standard polar coordinate shapes such as the sphere, cylinder, disk, torus or cone.

Example – The sphere.

Remember the sphere second mapping:
x = r * Cos(u) * Cos (v)
y = r * Sin(v)
z = r * Sin(u) * Cos (v)

if you replace the r parameter with the superformula in u, v or both you get the supershapes in 3D presented in Paul Bourke website. The formula becomes:

In u:
R(u) = second figure

And in v

R(v) = third figure

This gives us three variations of the supershape sphere formula:

U only:
x = R(u) * Cos(u) * Cos (v)
y = Sin(v)
z = R(u) * Sin(u) * Cos (v)

v only:
x = Cos(u) * R(v) * Cos (v)
y = R(v) * Sin(v)
z = Sin(u) * R(v) * Cos (v)

both u and v, which is the general representation seen on various websites and the basis of the Supershape_Sphere ICE Compound:
x = R(u) * Cos(u) * R(v) * Cos (v)
y = R(v) * Sin(v)
z = R(u) * Sin(u) * R(v) * Cos (v)

The parameters:

M is the angular coefficient multiplier. If m = 0 we get the sphere representation. If m is an integer, the shape is closed. If m is a fraction, the shape is open but we can rotate u or v multiple times to create a petal/star shape with intersecting polygons. M being a multiplier will control the number of points in the shape (wave or star like shape), example m=5 will create a shape with five bumps or start points.

a and b are the radius controls. They can be any numbers except zero (to avoid a division by zero error). Negative numbers need to be converted to positive using an absolute to avoid imaginary numbers issue.

n1, n2 and n3 are the exponent controlling the supershape. This is where the magic happen. They can be any numbers, positive or negatives. Small values have the greatest impact. Note that if you make n1 equal to zero, x, y and z will be equal to zero.

So google up "superformula" and "supershape", feed those parameters you found into the supershape formula, give it some colours and see how flexible the superformula is.

Cheers!

hey Daniel, any chance you could share the braided torus? i'm really interested into this shape, would be very nice!
thanks again for your awesome work on this

Max